Beta function (physics)

In quantum field theory, the beta function describes the dependence of a coupling constant on the energy scale resulting from the renormalization . The definition applies ${\displaystyle \beta (\alpha )}$ ${\displaystyle \alpha }$ ${\displaystyle \mu }$

${\displaystyle \beta (\alpha ):=\mu {\frac {\partial \alpha }{\partial \mu }}}$.

The sign of the beta function is of particular importance for the interaction associated with the corresponding coupling constant.

Examples

Quantum electrodynamics

The beta function of quantum electrodynamics is

${\displaystyle \beta (\alpha )={\frac {2\alpha ^{2}}{3\pi }}+{\mathcal {O}}(\alpha ^{3})~.}$

The positive sign means that the coupling is also small with small scales. The electromagnetic interaction thus becomes weaker with increasing distance.

Quantum chromodynamics

The beta function of quantum chromodynamics is

${\displaystyle \beta (\alpha _{s})=-\left(11-{\frac {2}{3}}n_{f}\right){\frac {\alpha _{s}^{2}}{2\pi }}+{\mathcal {O}}(\alpha _{s}^{3}),}$

where indicates the number of quark flavors. For the standard model of elementary particle physics , this leads to a negative sign. The decrease in the coupling constant as the scale rises is called asymptotic freedom . ${\displaystyle n_{f}}$ ${\displaystyle n_{f}=6}$

The contributions including the regulations are known ${\displaystyle {\mathcal {O}}(\alpha _{s}^{5})}$

literature

• Michael E. Peskin, Daniel V. Schroeder: An Introduction to Quantum Field Theory . Ed .: Perseus Books. Addison-Wesley, Reading, Massachusetts 1997, ISBN 978-0-201-50397-5 (English, fulviofrisone.com [PDF; 37.5 MB ]).

Individual evidence

1. ↑ Timo van Ritbergen, JAM Vermaseren, SA Larin: The Four loop beta function in quantum chromodynamics . High Energy Physics - Phenomenology. Ed .: CERN Document Server. January 27, 1997, doi : 10.1016 / S0370-2693 (97) 00370-5 (English, cern.ch [PDF; 266 kB ]).